Finite-time blow-up prevention by logistic source in parabolic-elliptic chemotaxis models with singular sensitivity in any dimensional setting

Abstract

In recent years, a lot of attention has been drawn to the question of whether logistic kinetics is sufficient to enforce the global existence of classical solutions or to prevent finite-time blow-up in various chemotaxis models. The current paper is to study the above question for the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source in any space dimensional setting, \begin{equation} \begin{cases} u_t=\Delta u-\chi\nabla\cdot (\frac{u}{v} \nabla v)+u(a(x,t)-b(x,t) u^{1+\sigma}),\quad &x\in \Omega\cr 0=\Delta v-\mu v+\nu u,\quad &x\in \Omega \quad \cr\frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0,\quad &x\in\partial\Omega, \end{cases} \end{equation} where $\Omega \subset \mathbb{R}^n$ is a bounded domain with smooth boundary $\partial\Omega$, $\chi$ is the singular chemotaxis sensitivity coefficient, $a(x,t)$ and $b(x,t)$ are positive smooth functions, $\mu,\nu$ are positive constants, and $\sigma\ge 0$. When $\sigma>0$, we prove that, for every given nonnegative initial data $0\not\equiv u_0\in C^0(\bar \Omega)$, (0.1) has a unique globally defined classical solution $(u_\sigma(x,t;u_0),v_\sigma(x,t;u_0))$ with $u_\sigma(x,0;u_0)=u_0(x)$, which shows that, in any space dimensional setting, strong logistic kinetics is sufficient to enforce the global existence of classical solutions and hence prevents the occurrence of finite-time blow-up even for arbitrarily large $\chi$. In addition, the solutions are shown to be uniformly bounded under the conditions \begin{equation*} a_{\inf}> \begin{cases} \frac{\mu \chi^2}{4}, &\text{if $0< \chi \leq 2,$}\\ \mu(\chi-1), &\text{if $\chi>2$.}\\ \end{cases} \end{equation*} When $\sigma=0$, we show that the classical solution $(u(x,t;u_0,0),v(x,t;u_0,0))$ exists globally and stays bounded provided that both $a(x,t)$ and $u_0(x)$ are not small.